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We consider within a finite element approach the usage of differentadaptively refined meshes for different variables in systems of nonlinear,time-depended PDEs. To resolve different solution behaviours of thesevariables, the meshes can be independently adapted. The resulting linearsystems are usually much smaller, when compared to the usage of a single mesh,and the overall computational runtime can be more than halved in such cases.Our multi-mesh method works for Lagrange finite elements of arbitrary degreeand is independent of the spatial dimension. The approach is well defined, andcan be implemented in existing adaptive finite element codes with minimaleffort. We show computational examples in 2D and 3D ranging from dendriticgrowth to solid-solid phase-transitions. A further application comes from fluiddynamics where we demonstrate the applicability of the approach for solving theincompressible Navier-Stokes equations with Lagrange finite elements of thesame order for velocity and pressure. The approach thus provides an easy toimplement alternative to stabilized finite element schemes, if Lagrange finiteelements of the same order are required.

A multi-mesh finite element method for Lagrange elements of arbitrary degree